Subgroup ($H$) information
| Description: | $C_{70}$ |
| Order: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Index: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Generators: |
$b^{1155}, b^{660}, b^{1386}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{11}\times D_{210}$ |
| Order: | \(4620\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Exponent: | \(2310\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $S_3\times C_{11}$ |
| Order: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Automorphism Group: | $S_3\times C_{10}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5:F_5^2$, of order \(100800\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4200\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $-3$ |
| Projective image | $C_{11}\times D_{105}$ |